§5 · Agent 5 · 𝔸ⁿ: lattices · 2026-10-18T18:42:13Z

Theorem 5.5 · $\mathbb A^n$: $\mathcal M_I=\{S_k\}$, so $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$

proved ★ key step

The summand criterion applies to $\mathbb A^n$ with every hypothesis checked, and the summand set is everything because each simple object is a shifted length-zero translate of $\mathrm{IC}_0$. This is a second proof of $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$: it uses no standard objects, no dévissage, no Proposition III or IV.1, and no decomposition theorem.

Theorem 5.5.

For $X=\mathbb A^n$ under $G=\mathrm{GL}_n$, every $n\ge1$, over $k=\mathbb C$ (Betti) and over $k=\overline{\mathbb F}_q$ ($\ell$-adic):

(a) hypotheses (L) and (Hk) of Proposition 5.4 hold, with $c_{N,M}=nM$;

(b) $\mathcal M_I=\operatorname{Simple}(\mathcal P_I)=\{S_k=\underline k_{\tau^k\mathcal O^n}[-k]:k\in\mathbb Z\}$, and each $S_k$ is the one-term decomposition of $\mathrm{IC}_{\tau^k}\ast_I\mathrm{IC}_0=S_k[k]$;

(c) $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$; more precisely $$\mathcal C_I=\operatorname{Tri}\{\delta_{\tau^k}\ast_I\mathrm{IC}_0:\ k\in\mathbb Z\},$$ so the skyscrapers at $\Omega=\langle\tau\rangle$ already generate, by extensions and shifts alone;

(d) (DT)(DT) · Decomposition: every generator IC_w ∗_I IC_0 is, equivariantly, a finite direct sum of shifted simple objects of C_I. Holds over F̄_q from a finite-type proper model of the convolution (purity and the stack decomposition theorem). is not used; it holds anyway (Sun, Thm 1.2 over $\overline{\mathbb F}q$; BBD 6.2.5 over $\mathbb C$), and then all of (a)–(c) of Proposition 5.4 hold verbatim, with explicit summands for $w\in\Omega$, $w=t^\lambda$, $w=w{\max}$ and the simple reflections (Computation 5.3).

Proof.

  1. (L). The level presentation is $Y_N=t^{-N}\mathcal O^n$, $Y_{N,M}=V_{N,M}$, $I_{N,M}=1+t^{N+M}M_n(\mathcal O)$; $H_{N,M}$ is connected with finitely many ($n(N+M)+1$) orbits, all stabilizers connected (Lemma 5.1(d)); the transitions are the smooth surjection $V_{N,M+1}\to V_{N,M}$ with fibres $\mathbb A^n$ and the closed embedding $V_{N,M}\hookrightarrow V_{N+1,M}$, t-exact for $c_{N,M}=nM$ (Lemma 5.2(a)). In the setting $\mathcal C_I$ is defined as the union of the finite levels, so the first half of I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III). is not an assumption here.
  1. (Hk). General (Proposition 5.4, last paragraph); only $\delta_{\tau^k}\in\mathcal H_{c,I}$ is used below.
  1. (b). By Computation 5.3(a), $\mathrm{IC}_{\tau^k}\ast\mathrm{IC}_0\simeq S_k[k]$, a direct sum with one term, so $S_k\in\mathcal M_I$ for every $k$; by Lemma 5.2(b) these are all the simple objects.
  1. (c). Proposition 5.4(c), direction $\Leftarrow$, which uses only (L), gives $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$. Directly: by Lemma 5.2(c) every object of $\mathcal C_I$ is an iterated extension of shifts of the $S_k=(\delta_{\tau^k}\ast\mathrm{IC}_0)[-k]$.
  1. (d). Nothing above invokes (DT)(DT) · Decomposition: every generator IC_w ∗_I IC_0 is, equivariantly, a finite direct sum of shifted simple objects of C_I. Holds over F̄_q from a finite-type proper model of the convolution (purity and the stack decomposition theorem).; the splittings that appear in Computation 5.3 are proved by parity.

Comparison (what is different from IV.1, which is not used). IV.1 verifies I1I1 · Finite levels: LX has a level presentation in which every object of C_I is pulled back from a finite level Y_{N,M} and each level has finitely many pieces (Iwahori-generation pages, Proposition III).–I3I3 · Triangularity on pieces: for every piece some Hecke translate of IC_0 or of a lower standard object is supported on its closure with nonzero constant restriction (Iwahori-generation pages). and runs the dévissage of Proposition III: standard objects $\Delta_P$ of pieces, the triangles $\Delta_{O_{m,j}}\to\underline k_{\Lambda_{m,j}}\to\underline k_{\Lambda_{m,j-1}}$, and the non-simple Hecke objects $\Delta_{t^{\lambda_{m,j}}}$, whose translates carry the shift $-2j(n-j)$. Here the t-structure of the colimit and its simple objects are identified (Lemma 5.2), the generators are simple objects $\delta_{\tau^k}$ of $\mathcal H_{c,I}$ supported on length-zero cells, and the conclusion comes from the summand criterion. The orbit facts are reproved in Lemma 5.1, so the argument does not depend on IV.1 or on Proposition III. Over both fields, by the same proof.

The point 0. Every object used is pulled back from a finite level; the point $0$ appears only inside the tails $t^M\mathcal O^n$, which are supports of the simple objects $S_{nM}$ (Lemma 5.2(d)).

n = 1. $I=K$, $\Omega=\widetilde W=\langle t\rangle$, $\tau=t$, $S_k=\underline k_{t^k\mathcal O}[-k]$: the statement reduces to spherical generation for $\mathbb A^1$ under $\mathbb G_m$.

Builds on

  • Lemma 5.1 · Orbits of $I$ on $L\mathbb A^n$: closures $\tau^k\mathcal O^n$, $\tau\in\Omega$
  • Lemma 5.2 · Simple objects of $\mathcal C_I$: $S_k=\underline k_{\tau^k\mathcal O^n}[-k]$
  • Computation 5.3 · $\mathrm{IC}_{\tau^k}\ast_I\mathrm{IC}_0=S_k[k]$ and the simple reflections
  • Proposition 5.4 · (I$_I$) The Iwahori summand criterion: generation iff $\mathcal M_I$ is all

Built on by

  • Summary 5.9 · Verdict: for $\mathbb A^n$ both Iwahori-level statements hold
  • Audit 8.9 · Audit of $\mathcal C_I^{\rm Iwa}=\mathcal C_I$ for $\mathbb A^n$ (A5-3, A5-5, A2-7)
  • Strategy 10.1 · Registry after round one: every target has a candidate proof; three adversarial audits are launched
  • Summary 10.2 · Verdict: the criterion and the bridge hold; the decided rows of Table 1 agree; GL₂/T fails at both levels
Original record, as the agent wrote it at 18:42 UTC

$\mathbb A^n$: $\mathcal M_I=\{S_k\}$, so $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$

Proof.

  1. (L). The level presentation is $Y_N=t^{-N}\mathcal O^n$, $Y_{N,M}=V_{N,M}$, $I_{N,M}=1+t^{N+M}M_n(\mathcal O)$; $H_{N,M}$ is connected with finitely many ($n(N+M)+1$) orbits, all stabilizers connected (Lemma 5.1(d)); the transitions are the smooth surjection $V_{N,M+1}\to V_{N,M}$ with fibres $\mathbb A^n$ and the closed embedding $V_{N,M}\hookrightarrow V_{N+1,M}$, t-exact for $c_{N,M}=nM$ (Lemma 5.2(a)). In the setting $\mathcal C_I$ is defined as the union of the finite levels, so the first half of I1 is not an assumption here.
  1. (Hk). General (Proposition 5.4, last paragraph); only $\delta_{\tau^k}\in\mathcal H_{c,I}$ is used below.
  1. (b). By Computation 5.3(a), $\mathrm{IC}_{\tau^k}\ast\mathrm{IC}_0\simeq S_k[k]$, a direct sum with one term, so $S_k\in\mathcal M_I$ for every $k$; by Lemma 5.2(b) these are all the simple objects.
  1. (c). Proposition 5.4(c), direction $\Leftarrow$, which uses only (L), gives $\mathcal C_I^{\mathrm{Iwa}}=\mathcal C_I$. Directly: by Lemma 5.2(c) every object of $\mathcal C_I$ is an iterated extension of shifts of the $S_k=(\delta_{\tau^k}\ast\mathrm{IC}_0)[-k]$.
  1. (d). Nothing above invokes (DT); the splittings that appear in Computation 5.3 are proved by parity.

Comparison (what is different from IV.1, which is not used). IV.1 verifies I1–I3 and runs the dévissage of Proposition III: standard objects $\Delta_P$ of pieces, the triangles $\Delta_{O_{m,j}}\to\underline k_{\Lambda_{m,j}}\to\underline k_{\Lambda_{m,j-1}}$, and the non-simple Hecke objects $\Delta_{t^{\lambda_{m,j}}}$, whose translates carry the shift $-2j(n-j)$. Here the t-structure of the colimit and its simple objects are identified (Lemma 5.2), the generators are simple objects $\delta_{\tau^k}$ of $\mathcal H_{c,I}$ supported on length-zero cells, and the conclusion comes from the summand criterion. The orbit facts are reproved in Lemma 5.1, so the argument does not depend on IV.1 or on Proposition III. Over both fields, by the same proof.

The point 0. Every object used is pulled back from a finite level; the point $0$ appears only inside the tails $t^M\mathcal O^n$, which are supports of the simple objects $S_{nM}$ (Lemma 5.2(d)).

n = 1. $I=K$, $\Omega=\widetilde W=\langle t\rangle$, $\tau=t$, $S_k=\underline k_{t^k\mathcal O}[-k]$: the statement reduces to spherical generation for $\mathbb A^1$ under $\mathbb G_m$.

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